The \emph{generalized sorting problem} is a restricted version of standard comparison sorting where we wish to sort $n$ elements but only a subset of pairs are allowed to be compared. Formally, there is some known graph $G = (V, E)$ on the $n$ elements $v_1, \dots, v_n$, and the goal is to determine the true order of the elements using as few comparisons as possible, where all comparisons $(v_i, v_j)$ must be edges in $E$. We are promised that if the true ordering is $x_1 < x_2 < \cdots < x_n$ for $\{x_i\}$ an unknown permutation of the vertices $\{v_i\}$, then $(x_i, x_{i+1}) \in E$ for all $i$: this Hamiltonian path ensures that sorting is actually possible. In this work, we improve the bounds for generalized sorting on both random graphs and worst-case graphs. For Erd\H{o}s-Renyi random graphs $G(n, p)$ (with the promised Hamiltonian path added to ensure sorting is possible), we provide an algorithm for generalized sorting with an expected $O(n \log (np))$ comparisons, which we prove to be optimal for query complexity. This strongly improves over the best known algorithm of Huang, Kannan, and Khanna (FOCS 2011), which uses $\tilde{O}(\min(n \sqrt{np}, n/p^2))$ comparisons. For arbitrary graphs $G$ with $n$ vertices and $m$ edges (again with the promised Hamiltonian path), we provide an algorithm for generalized sorting with $\tilde{O}(\sqrt{mn})$ comparisons. This improves over the best known algorithm of Huang et al., which uses $\min(m, \tilde{O}(n^{3/2}))$ comparisons.
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This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the fractional Brownian motion and the singular kernel, which is overcome by means of the Malliavin calculus and fine estimates of several multiple singular integrals. Consequently, these two methods are proved to be strongly convergent with order nearly $\min\{2(H+\alpha-1), \alpha\}$, where $H \in (1/2,1)$ and $\alpha\in(1-H,1)$ respectively characterize the singularity levels of fractional noises and singular kernels in the underlying equation. This result improves the existing convergence order $H+\alpha-1$ of Euler methods for the nonlinear case, and gives a positive answer to the open problem raised in [4]. As an application of the theoretical findings, we further investigate the complexity of the multilevel Monte Carlo simulation based on the fast Euler method, which turns out to behave better performance than the standard Monte Carlo simulation when computing the expectation of functionals of the considered equation.
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, biharmonic and Stokes equations. These equations depend on a parameter, denoted $\alpha$, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter $\alpha$, at an increased cost that scales as $\log\alpha$. Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible.
Time-varying parameter VARs with stochastic volatility are routinely used for structural analysis and forecasting in settings involving a few endogenous variables. Applying these models to high-dimensional datasets has proved to be challenging due to intensive computations and over-parameterization concerns. We develop an efficient Bayesian sparsification method for a class of models we call hybrid TVP-VARs--VARs with time-varying parameters in some equations but constant coefficients in others. Specifically, for each equation, the new method automatically decides whether the VAR coefficients and contemporaneous relations among variables are constant or time-varying. Using US datasets of various dimensions, we find evidence that the parameters in some, but not all, equations are time varying. The large hybrid TVP-VAR also forecasts better than many standard benchmarks.
In this paper, we revisit the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) and provide excess population risks for some special classes of functions that are faster than the previous results of general convex and strongly convex functions. In the first part of the paper, we study the case where the population risk function satisfies the Tysbakov Noise Condition (TNC) with some parameter $\theta>1$. Specifically, we first show that under some mild assumptions on the loss functions, there is an algorithm whose output could achieve an upper bound of $\tilde{O}((\frac{1}{\sqrt{n}}+\frac{\sqrt{d\log \frac{1}{\delta}}}{n\epsilon})^\frac{\theta}{\theta-1})$ for $(\epsilon, \delta)$-DP when $\theta\geq 2$, here $n$ is the sample size and $d$ is the dimension of the space. Then we address the inefficiency issue, improve the upper bounds by $\text{Poly}(\log n)$ factors and extend to the case where $\theta\geq \bar{\theta}>1$ for some known $\bar{\theta}$. Next we show that the excess population risk of population functions satisfying TNC with parameter $\theta\geq 2$ is always lower bounded by $\Omega((\frac{d}{n\epsilon})^\frac{\theta}{\theta-1}) $ and $\Omega((\frac{\sqrt{d\log \frac{1}{\delta}}}{n\epsilon})^\frac{\theta}{\theta-1})$ for $\epsilon$-DP and $(\epsilon, \delta)$-DP, respectively. In the second part, we focus on a special case where the population risk function is strongly convex. Unlike the previous studies, here we assume the loss function is {\em non-negative} and {\em the optimal value of population risk is sufficiently small}. With these additional assumptions, we propose a new method whose output could achieve an upper bound of $O(\frac{d\log\frac{1}{\delta}}{n^2\epsilon^2}+\frac{1}{n^{\tau}})$ for any $\tau\geq 1$ in $(\epsilon,\delta)$-DP model if the sample size $n$ is sufficiently large.
The problem of $d$-Path Vertex Cover, $d$-PVC lies in determining a subset $F$ of vertices of a given graph $G=(V,E)$ such that $G \setminus F$ does not contain a path on $d$ vertices. The paths we aim to cover need not to be induced. It is known that the $d$-PVC problem is NP-complete for any $d \ge 2$. When parameterized by the size of the solution $k$, 5-PVC has direct trivial algorithm with $\mathcal{O}(5^kn^{\mathcal{O}(1)})$ running time and, since $d$-PVC is a special case of $d$-Hitting Set, an algorithm running in $\mathcal{O}(4.0755^kn^{\mathcal{O}(1)})$ time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in $\mathcal{O}(4^kn^{\mathcal{O}(1)})$ time.
This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation, and the computational domain is divided into overlapping sub-domains centered on each node. Combining with moving least square approximation and local Taylor expansion, derivatives of oil-phase pressure at the central node are approximated by a generalized difference operator in the local subdomain. By introducing the first-order upwind scheme of phase permeability, and combining the discrete boundary conditions, fully implicit GFDM discrete nonlinear equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation technology, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, and points out the significant effect of the symmetry or uniformity of the node allocation in the node influence domain on the accuracy of the generalized difference operator, and the radius of node influence domain should be as small as possible to achieve high calculation accuracy, which is a significant difference between the studied parabolic two-phase porous flow problem and the elliptic equation previously studied by GFDM. In all, the upwind GFDM with the fully implicit nonlinear solver and related analysis about computational performances given in this work may provide a critical reference for developing a general-purpose meshless numerical simulator for porous flow problems.
We show that a specific skew-symmetric form of hyperbolic problems leads to energy conservation and an energy bound. Next, the compressible Euler equations is transformed to this skew-symmetric form and it is explained how to obtain an energy estimate. Finally we show that the new formulation lead to energy stable and energy conserving discrete approximations if the scheme is formulated on summation-by-parts form.
Hamilton and Moitra (2021) showed that, in certain regimes, it is not possible to accelerate Riemannian gradient descent in the hyperbolic plane if we restrict ourselves to algorithms which make queries in a (large) bounded domain and which receive gradients and function values corrupted by a (small) amount of noise. We show that acceleration remains unachievable for any deterministic algorithm which receives exact gradient and function-value information (unbounded queries, no noise). Our results hold for the classes of strongly and nonstrongly geodesically convex functions, and for a large class of Hadamard manifolds including hyperbolic spaces and the symmetric space $\mathrm{SL}(n) / \mathrm{SO}(n)$ of positive definite $n \times n$ matrices of determinant one. This cements a surprising gap between the complexity of convex optimization and geodesically convex optimization: for hyperbolic spaces, Riemannian gradient descent is optimal on the class of smooth and and strongly geodesically convex functions, in the regime where the condition number scales with the radius of the optimization domain. The key idea for proving the lower bound consists of perturbing the hard functions of Hamilton and Moitra (2021) with sums of bump functions chosen by a resisting oracle.
In warehouses, order picking is known to be the most labor-intensive and costly task in which the employees account for a large part of the warehouse performance. Hence, many approaches exist, that optimize the order picking process based on diverse economic criteria. However, most of these approaches focus on a single economic objective at once and disregard ergonomic criteria in their optimization. Further, the influence of the placement of the items to be picked is underestimated and accordingly, too little attention is paid to the interdependence of these two problems. In this work, we aim at optimizing the storage assignment and the order picking problem within mezzanine warehouse with regards to their reciprocal influence. We propose a customized version of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) for optimizing the storage assignment problem as well as an Ant Colony Optimization (ACO) algorithm for optimizing the order picking problem. Both algorithms incorporate multiple economic and ergonomic constraints simultaneously. Furthermore, the algorithms incorporate knowledge about the interdependence between both problems, aiming to improve the overall warehouse performance. Our evaluation results show that our proposed algorithms return better storage assignments and order pick routes compared to commonly used techniques for the following quality indicators for comparing Pareto fronts: Coverage, Generational Distance, Euclidian Distance, Pareto Front Size, and Inverted Generational Distance. Additionally, the evaluation regarding the interaction of both algorithms shows a better performance when combining both proposed algorithms.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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